Hi-
Brian Postow asked:
>I've been wondering this for a while, and since someone else just
>asked me, I though I'd ask here. Does anyone know the origin of the notation
>\Sigma_i and \Pi_i for the different levels of the arithmetic
>hierarchy?
I don't know about the arithmetical hierarchy, but Pi and Sigma were
linked to quantification at a very early stage. For example, see page 195 of:
C.S. Peirce, On the algebra of logic, American Journal of Mathematics,
Volume 7, No. 2 (Jan. 1885), 180-196.
If you have access to JSTOR, you can download this from:
http://links.jstor.org/sici?sici=0002-9327%28188501%297%3A2%3C180%3AOTAOLA%3E2.0.CO%3B2-2
According to the historical comments in Church's Introduction to
Mathematical Logic (see section 49), Peirce credits O.H. Mitchell with
the notation, but the use of the "operator variable" (which allows
alternation of quantifiers) is due to Peirce, as is the use of the word
"quantifier."
Church also notes that quantifiers were used earlier in
Frege's Begriffsschrift of 1879.
Since predicates were treated by some early authors as boolean
valued functions, quantification naturally was linked to products
and sums.
-Jeff
--
Jeff Hirst jlh at math.appstate.edu
Professor of Mathematics
Appalachian State University, Boone, NC 28608
vox:828-262-2861 fax:828-265-8617